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T-Duality
T-Duality is a duality, or equivalence between two String Theoryies. In contrast to S-Duality, T-Duality is a purely stringy concept. T-Duality equates a String Theory compactified around a distance R with another compactified around a distance of \frac{\alpha'}{R}=\frac{\ell_s^2}{R} . T-Duality for Bosonic Strings Beginning with a simple toy model, Bosonic String Theory, we compactify a spatial dimension, say x^9 (9 is randomly chosen as Bosonic String Theory is 26-dimensional, not 10-dimensional), such that: . x^9\sim x^9+2\pi R The ground Wavefunction is e^{ip_0^9x^9/\hbar} . It is clear that this is single-valued only when p_0^9=\hbar \frac nR There fore, the momenta is quantised by the above equation. Then, \alpha_0^\mu=\tilde\alpha_0^\mu= \frac{\ell_s}{\hbar}\frac nR \alpha_0^\mu+\tilde\alpha_0^\mu= \frac{2\ell_s}{\hbar} \frac nR = However, a Closed String can actually wrap as many times around a circle as it wishes, not necessarily once. This "as many times" is called the winding number w Then, x^9\sim x^9+2\pi R \alpha_0^\mu-\tilde\alpha_0^\mu= \frac{wR}{\ell_s} When w=0 (uncompactified), the RHS becomes 0. If we consider the momentum, it is still $ p=\frac nR $, but the left- and right- moving momenta are: p_-= \hbar\left(\frac nR - \frac1{\ell_s^2} wR \right) p_+= \hbar\left(\frac nR + \frac1{\ell_s^2} wR \right) The mass spectrum is also intuitively modified as (which is clear from the relation between the mass and the momenta); : $$ m=\frac{2\pi T\ell_s}{c_0^2} \sqrt{N+\tilde N-a-\tilde a + \ell_s^2 \frac{n^2}{R^2} \frac{1}{\ell_s^2}w^2R^2 } $$ If we talke the limit as R\to\infty , w\to0 and if we take the limit as R\to0 , then n\to0 , and the compactified dimension reappers. This is explained by the following transformations between the winding number and the momentum quantisation number; and between the compactification radii. w\leftrightarrow n R\leftrightarrow \frac{ell_s^2}{R} These transformations are called T-Duality. This is also equivalent to negating the right-moving mode of oscillation; and therefore, the field X^\mu . Considering T-Duality for Open Strings, we see that a normal derivative becomes a tangential derivative, therefore exchanging Dirchilet and Newmann Boundary conditions. This immediately makes the existence of D-Branes necessary. T-Duality for Type II Strings Most of the ideas of the previous section carry over to Type II (Type IIA and Type IIB) Strings; however there is an additional result. In the previous section, we learnt that T-Duality negates the bosonic field X^\mu . By the manifest Worldsheet Supersymmetry of RNS String Theory, this also implies that the fermionic field \psi ^\mu is negated. This immediately implies that the GSO Projection also flips sign: \operatorname{T}: \mathcal P^-_\operatorname{GSO}\leftrightarrow \mathcal P^+_\operatorname{GSO} As the Type IIA String Theory and Type IIB String Theory differ only by the GSO Projection, this means that T-Duality exchanges Type IIA String Theory and Type IIB String Theory. T-Duality for Type H Strings Type H String Theory, or Heterotic String Theory, is also affected by T-Duality. The weight lattice of \frac{\operatorname{Spin}\left(32\right)}{\mathbb{Z}_2} (the gauge group of the Type HO String Theory is given by \Gamma^{16} while the weight lattice of E(8)\times E(8) is given by \Gamma^8\oplus\Gamma^8 . Since \Gamma^{8}\oplus \Gamma^8\oplus\Gamma^{1,1}= \Gamma^{16} \oplus\Gamma^{1,1} , it follows that Type HO String Theory is T-Dual to Type HE String Theory. Category:String Theory